Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle AOB = 3x - 22$, and $ m \angle BOC = 8x - 108$, find $m\angle AOB$. $O$ $A$ $C$ $B$
Explanation: From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {3x - 22} + {8x - 108} = {90}$ Combine like terms: $ 11x - 130 = 90$ Add $130$ to both sides: $ 11x = 220$ Divide both sides by $11$ to find $x$ $ x = 20$ Substitute $20$ for $x$ in the expression that was given for $m\angle AOB$ $ m\angle AOB = 3({20}) - 22$ Simplify: $ {m\angle AOB = 60 - 22}$ So ${m\angle AOB = 38}$.